Editing Polycube (section)

==Enumerating polycubes==
[[image:AGK-pentacube.png|thumb|right|A [[Chirality (mathematics)|chiral]] pentacube]]

Like [[polyomino]]es, polycubes can be enumerated in two ways, depending on whether [[Chirality (mathematics)|chiral]] pairs of polycubes (those equivalent by [[Reflection symmetry|mirror reflection]], but not by using only translations and rotations) are counted as one polycube or two. For example, 6 tetracubes are achiral and one is chiral, giving a count of 7 or 8 tetracubes respectively.<ref name=symmetry>{{citation|last=Lunnon |first=W. F. |contribution=Symmetry of Cubical and General Polyominoes |editor-last=Read |editor-first=Ronald C. |title=Graph Theory and Computing |place=New York |publisher=Academic Press |year=1972 |pages=101–108 |isbn=978-1-48325-512-5}}</ref> Unlike polyominoes, polycubes are usually counted with mirror pairs distinguished, because one cannot turn a polycube over to reflect it as one can a polyomino given three dimensions.  In particular, the [[Soma cube]] uses both forms of the chiral tetracube.

Polycubes are classified according to how many cubical cells they have:<ref>[http://recmath.org/PolyPages/PolyPages/index.htm?Polycubes.html Polycubes, at The Poly Pages]</ref>

{| class=wikitable
|-
!''n''
!Name of ''n''-polycube
!Number of one-sided ''n''-polycubes<br>(reflections counted as distinct)<br>{{OEIS|id=A000162}}
!Number of free ''n''-polycubes<br>(reflections counted together)<br>{{OEIS|id=A038119}}
|-
|1
|monocube
|align=right|1
|align=right|1
|-
|2
|dicube
|align=right|1
|align=right|1
|-
|3
|tricube
|align=right|2
|align=right|2
|-
|4
|tetracube
|align=right|8
|align=right|7
|-
|5
|pentacube
|align=right|29
|align=right|23
|-
|6
|hexacube
|align=right|166
|align=right|112
|-
|7
|heptacube
|align=right|1023
|align=right|607
|-
|8
|octacube
|align=right|6922
|align=right|3811
|}

Fixed polycubes (both reflections and rotations counted as distinct {{OEIS|id=A001931}}), one-sided polycubes, and free polycubes have been enumerated up to ''n''=22. Specific families of polycubes have also been investigated.<ref>[http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p26/pdf "Enumeration of Specific Classes of Polycubes", Jean-Marc Champarnaud et al, Université de Rouen, France] PDF</ref><ref>[https://arxiv.org/abs/1311.4836 "Dirichlet convolution and enumeration of pyramid polycubes", C. Carré, N. Debroux, M. Deneufchâtel, J. Dubernard, C. Hillairet, J. Luque, O. Mallet; November 19, 2013] PDF</ref>